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Astron. Astrophys. 341, 902-911 (1999)

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4. Behavior of the continuum polarization

We applied the computer code to the nine different model atmospheres introduced in Sect. 2.4. After a presentation of the resulting continuum polarization, we identify the reasons for the wavelength dependence (Sect. 4.1) and for the differences between the various model atmospheres (Sect. 4.2). The scattering coefficient and the temperature gradient turn out to be the most important physical quantities.

Fig. 4 presents the calculated continuum polarization for different model atmospheres as a function of µ (left panel) and wavelength (right panel). Let us now summarize the most significant results:

  • The CLV is largely determined by simple geometry since Rayleigh and Thomson scattering act as dipole scattering (cf. Sect. 5.1). Due to axial symmetry the scattering polarization vanishes at disk center for all models.

  • With increasing wavelength the polarization decreases steeply, mainly due to the wavelength dependence of the Rayleigh scattering. In Sect. 4.1.2 we will show a further effect due to the wavelength dependence of the Planck function.

  • Within each of the model groups {AYFLUXT1, AYP2, AYCOOL8} and {FALP3, FALF4, FALC5, FALA7} the polarization is smaller for hotter atmospheres.

  • For the two model atmospheres with no chromosphere, AYCOOL8 and AND9, the polarizations are similar to those of the other models. Thus the chromosphere does not seem to be very important for the formation of the polarization. AYFLUXT1 and AYP2 are the exceptions to this rule and have small contributions in the chromosphere, as will be shown in Sect. 4.2.

  • The two average quiet Sun model atmospheres, FALC5 and MACKKL6, differ significantly only in the upper chromosphere. However, their polarizations are almost identical, which again demonstrates the low relevance of the upper chromosphere.

[FIGURE] Fig. 4. Center-to-limb variation (left ) and wavelength dependence (right ) of the continuum polarization for a representative set of solar model atmospheres. The curves of the other model atmospheres lie between the plotted curves.

4.1. Wavelength dependence

This section is devoted to a qualitative study of the wavelength dependence of the continuum polarization. The essential points are summarized in Fig. 5. The results obtained below are valid for all models. The following discussion is divided into two parts corresponding to the two most important physical quantities, namely the scattering coefficient and the temperature gradient.

[FIGURE] Fig. 5a-f. These plots clarify the causes of the wavelength dependence of the continuum polarization (panel a ). The two influencing quantities are the scattering coefficient, the logarithm of which is displayed in panels b and e , and the limb darkening at the height where the contribution function of Stokes Q has a maximum (panels c and f ). Shown are the results for two representative model atmospheres, AYFLUXT1 and MACKKL6. In panel d the ratio between two Planck functions with different temperatures is plotted to explain the wavelength dependence of the limb darkening (see text).

4.1.1. Scattering coefficient

Between 4000 Å and 8000 Å the scattering coefficient decreases by approximately a factor of ten in the photosphere, as shown in panels b and e of Fig. 5. The wavelength dependence of the scattering coefficient comes from the Rayleigh scattering. A smaller scattering coefficient results in a smaller number of scattering processes per unit volume and therefore in a lower polarization. Furthermore, the difference in [FORMULA] is larger between 4000 Å and 6000 Å than between 6000 Å and 8000 Å, which is well reflected by the steeper decline of the polarization at smaller wavelengths.

4.1.2. Temperature gradient

The temperature gradient is directly responsible for limb darkening. Panels c and f of Fig. 5 show the CLV of the intensity at the height in the atmosphere where the contribution function (see Sect. 4.2.1) of Stokes Q has a maximum. At this height, which is wavelength and model dependent, the limb darkening is most relevant for the formation of the polarization. This would not be true at the top of the atmosphere, because the formation heights of Stokes I and Q overlap (cf. Fig. 6).

[FIGURE] Fig. 6a-f. Illustration of various factors contributing to the differences in the continuum polarization between different model atmospheres: a : CLV of the continuum polarization at 6000 Å at the top of the atmosphere. b : Temperature as a function of geometric height. c : CLV of the continuum intensity relative to the intensity at disk center at the height where [FORMULA] reaches the maximum at 6000 Å. d : Logarithm of the scattering coefficient at 6000 Å. e : Contribution function of Stokes I ([FORMULA], thin lines) and of Stokes Q ([FORMULA], thick lines) at 6000 Å and [FORMULA]. f : Relative scattering coefficient at 6000 Å. Description of lines: solid : AYFLUXT1; dotted : FALF4; dashed : FALA7; dashed-dotted : AND9.

The greater the limb darkening the more anisotropic is the radiation field, and the greater the polarization produced. Fig. 5 clearly shows that the limb darkening decreases with increasing wavelength. This enhances the effect that the scattering coefficient yields a smaller polarization at larger wavelengths.

It is interesting to note the fact that the wavelength dependence of the limb darkening can at least partially be explained by properties of the Planck function, as was pointed out to us by S.K. Solanki (private communication). For simplicity, we assume the absorption coefficient to be wavelength independent and the continuum intensity to be black-body radiation. We consider the Planck function [FORMULA] and fix two temperatures [FORMULA] and [FORMULA] with [FORMULA]. The ratio between two Planck functions, one at temperature [FORMULA], the other at [FORMULA], is given by


which has the asymptotic values


The ratio [FORMULA] is a monotonically decreasing function of wavelength if [FORMULA], as shown in panel d of Fig. 5 where [FORMULA] is plotted for two typical temperatures in the photosphere.

In a grey atmosphere the relation between temperature and optical depth is wavelength independent. Therefore the lower value of [FORMULA] at higher wavelengths corresponds to a less pronounced limb darkening, in agreement with panels c and f of Fig. 5. This in turn results in a decreasing polarization with wavelength, even in the case of a grey atmosphere.

4.2. Sources of model dependence

In this section the reasons for the model dependence of the continuum polarization are investigated. Due to the form of the total source function (5) it is natural to examine the influence of the relative scattering coefficient


This however turns out to be insignificant for explaining the model dependence of the continuum polarization. Rather we find that the temperature gradient in combination with the scattering coefficient, [FORMULA], appear to be most important, as demonstrated in Fig. 6. Let us now discuss the various quantities plotted in that figure.

4.2.1. Contribution functions

For diagnostic work it is useful to know the location in the atmosphere where the emerging radiation is produced. This information is contained in the contribution function [FORMULA]. To compare the effect of different solar model atmospheres on the polarization we introduce the contribution function with respect to the geometric height z, which is defined by the equation


The integration bounds in Eq. (14) are so chosen for formal convenience. However, the errors produced by integrating from minus to plus infinity, instead of only integrating over the atmospheric slab considered, are negligible.

Panel e of Fig. 6 displays the contribution functions of Stokes I, [FORMULA], and of Stokes Q, [FORMULA], at 6000 Å and [FORMULA]. [FORMULA] peaks around a geometric height of 100 km in all models, which shows that the continuum intensity is formed in the lower photosphere. The maximum of [FORMULA] lies higher but still in the photosphere for all models. The fact that the polarization is primarily formed in the photosphere explains the irrelevance of the missing chromosphere in the cool models, AYCOOL8 and AND9, and the equality of the polarizations of models FALC5 and MACKKL6. Only in the AYFLUXT1 and AYP2 atmospheres a relevant part of Stokes Q is produced in the chromosphere, which shows the importance of calculating the opacities in the non-LTE case.

Note that according to the definition (14) the contribution function [FORMULA] is proportional to [FORMULA] and not to [FORMULA]. Therefore, [FORMULA] is the relevant quantity when interpreting the contribution functions (cf. panels d and e in Fig. 6).

4.2.2. Relative scattering coefficient

In the photosphere the values of [FORMULA] are very similar in all nine model atmospheres (Fig. 6 panel f). Therefore [FORMULA] will not cause differences in the polarization between the atmospheres in which Stokes Q is formed in the photosphere. Although AYFLUXT1 and AYP2 have a much smaller [FORMULA] in the chromosphere, their emergent polarization is not correspondingly reduced. We conclude that [FORMULA] is insignificant for explaining the diversity of the continuum polarization in different model atmospheres.

4.2.3. Scattering coefficient

In the photosphere the scattering coefficients are also almost identical in all the model atmospheres and thus do not lead to a model dependence. Only in the cases of AYFLUXT1 and AYP2 the chromosphere must not be neglected. Despite the weaker limb darkening of AYFLUXT1 as compared to AND9, a significantly higher [FORMULA] in the chromosphere for the former model results in a correspondingly larger polarization (cf. Fig. 6).

4.2.4. Temperature gradient

According to panel e in Fig. 6 the intensity is produced in the photosphere for all model atmospheres. Because the intensity source function is in first approximation equal to the Planck function, the CLV of the continuum intensity is directly related to the temperature gradient, which is confirmed by inspection of panels b and c : the greater the temperature gradient in the photosphere (approximately between 0 and 500 km) the more pronounced is the limb darkening.

Let us now consider the model atmospheres FALF4, FALA7 and AND9. The differences in [FORMULA] and [FORMULA] are small. However, the limb-darkening curves are not identical and a greater CLV of the intensity corresponds to a higher degree of polarization for these atmospheres.

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© European Southern Observatory (ESO) 1999

Online publication: December 16, 1998